Fungrim entry: a21395

R_{-a}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = {A}^{-a} \sum_{N=0}^{\infty} \frac{\left(a\right)_{N}}{\left(n \beta\right)_{N}} T_{N}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[1 - \frac{z_{1}}{A}, 1 - \frac{z_{2}}{A}, \ldots, 1 - \frac{z_{n}}{A}\right]\right)\; \text{ where } A = \frac{1}{n} \sum_{k=1}^{n} z_{k}

Assumptions:

a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \beta \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \left(z_{k} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|1 - \frac{n z_{k}}{\sum_{j=1}^{n} z_{j}}\right| < 1 \;\text{ for all } k \in \{1, 2, \ldots, n\}\right) \;\mathbin{\operatorname{and}}\; \left(\operatorname{Re}\!\left(z_{k}\right) > 0 \;\text{ for all } k \in \{1, 2, \ldots, n\}\right)

TeX:

R_{-a}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = {A}^{-a} \sum_{N=0}^{\infty} \frac{\left(a\right)_{N}}{\left(n \beta\right)_{N}} T_{N}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[1 - \frac{z_{1}}{A}, 1 - \frac{z_{2}}{A}, \ldots, 1 - \frac{z_{n}}{A}\right]\right)\; \text{ where } A = \frac{1}{n} \sum_{k=1}^{n} z_{k}

a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \beta \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \left(z_{k} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|1 - \frac{n z_{k}}{\sum_{j=1}^{n} z_{j}}\right| < 1 \;\text{ for all } k \in \{1, 2, \ldots, n\}\right) \;\mathbin{\operatorname{and}}\; \left(\operatorname{Re}\!\left(z_{k}\right) > 0 \;\text{ for all } k \in \{1, 2, \ldots, n\}\right)

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonHypergeometricR`	$R_{-a}\!\left(b, z\right)$	Carlson multivariate hypergeometric function
`Pow`	${a}^{b}$	Power
`Sum`	$\sum_{n} f(n)$	Sum
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`CarlsonHypergeometricT`	$T_{N}\!\left(b, z\right)$	Term in expansion of Carlson multivariate hypergeometric function
`Infinity`	$\infty$	Positive infinity
`RR`	$\mathbb{R}$	Real numbers
`OpenInterval`	$\left(a, b\right)$	Open interval
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("a21395"),
    Formula(Where(Equal(CarlsonHypergeometricR(Neg(a), List(Repeat(beta, n)), List(z_(k), For(k, 1, n))), Mul(Pow(A, Neg(a)), Sum(Mul(Div(RisingFactorial(a, N), RisingFactorial(Mul(n, beta), N)), CarlsonHypergeometricT(N, List(Repeat(beta, n)), List(Sub(1, Div(z_(k), A)), For(k, 1, n)))), For(N, 0, Infinity)))), Def(A, Mul(Div(1, n), Sum(z_(k), For(k, 1, n)))))),
    Variables(a, beta, z_, n),
    Assumptions(And(Element(a, RR), Element(beta, OpenInterval(0, Infinity)), Element(n, ZZGreaterEqual(1)), All(And(Element(z_(k), CC), Less(Abs(Sub(1, Div(Mul(n, z_(k)), Sum(z_(j), For(j, 1, n))))), 1)), ForElement(k, Range(1, n))), All(Greater(Re(z_(k)), 0), ForElement(k, Range(1, n))))))

Topics using this entry

Series representations of Carlson symmetric elliptic integrals